# A Life in Games

John Horton Conway claims to have never worked a day in his life. This adaptation from the biography Genius at Play shows how serious advances such as the surreal numbers can spring out of fun and games.

Princeton University, Office of Communications, Denise Applewhite

John Horton Conway at Princeton University in 2009.

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Gnawing on his left index finger with his chipped old British teeth, temporal veins bulging and brow pensively squinched beneath the day-before-yesterday’s hair, the mathematician John Horton Conway unapologetically whiles away his hours tinkering and thinkering — which is to say he’s ruminating, although he will insist he’s doing nothing, being lazy, playing games.

Based at Princeton University, though he found fame at Cambridge (as a student and professor from 1957 to 1987), Conway, 77, claims never to have worked a day in his life. Instead, he purports to have frittered away reams and reams of time playing. Yet he is Princeton’s John von Neumann Professor in Applied and Computational Mathematics (now emeritus). He’s a fellow of the Royal Society. And he is roundly praised as a genius. “The word ‘genius’ gets misused an awful lot,” said Persi Diaconis, a mathematician at Stanford University. “John Conway is a genius. And the thing about John is he’ll think about anything.… He has a real sense of whimsy. You can’t put him in a mathematical box.”

The hoity-toity Princeton bubble seems like an incongruously grand home base for someone so gamesome. The campus buildings are Gothic and festooned with ivy. It’s a milieu where the well-groomed preppy aesthetic never seems passé. By contrast, Conway is rumpled, with an otherworldly mien, somewhere between The Hobbit’s Bilbo Baggins and Gandalf. Conway can usually be found loitering in the mathematics department’s third-floor common room. The department is housed in the 13-story Fine Hall, the tallest tower in Princeton, with Sprint and AT&T cell towers on the rooftop. Inside, the professor-to-undergrad ratio is nearly 1-to-1. With a querying student often at his side, Conway settles either on a cluster of couches in the main room or a window alcove just outside the fray in the hallway, furnished with two armchairs facing a blackboard — a very edifying nook. From there Conway, borrowing some Shakespeare, addresses a familiar visitor with his Liverpudlian lilt:

Welcome! It’s a poor place but mine own!

Conway’s contributions to the mathematical canon include innumerable games. He is perhaps most famous for inventing the Game of Life in the late 1960s. The Scientific American columnist Martin Gardner called it “Conway’s most famous brainchild.” This is not Life the family board game, but Life the cellular automaton. A cellular automaton is a little machine with groups of cells that evolve from iteration to iteration in discrete rather than continuous time — in seconds, say, each tick of the clock advances the next iteration, and over time, behaving a bit like a transformer or a shape-shifter, the cells evolve into something, anything, everything else. Life is played on a grid, like tic-tac-toe, where its proliferating cells resemble skittering microorganisms viewed under a microscope.

The Game of Life is not really a game, strictly speaking. Conway calls it a “no-player never-ending” game. The recording artist and composer Brian Eno once recalled that seeing an electronic Game of Life exhibit on display at the Exploratorium in San Francisco gave him a “shock to the intuition.” “The whole system is so transparent that there should be no surprises at all,” Eno said, “but in fact there are plenty: The complexity and ‘organic-ness’ of the evolution of the dot patterns completely beggars prediction.” And as suggested by the narrator in an episode of the television show Stephen Hawking’s Grand Design, “It’s possible to imagine that something like the Game of Life, with only a few basic laws, might produce highly complex features, perhaps even intelligence. It might take a grid with many billions of squares, but that’s not surprising. We have many hundreds of billions of cells in our brains.”

Life was among the first cellular automata and remains perhaps the best known. It was coopted by Google for one of its Easter eggs: Type in “Conway’s Game of Life,” and alongside the search results ghostly light-blue cells will appear and gradually overrun the page. Practically speaking, the game nudged cellular automata and agent-based simulations into use in the complexity sciences, where they model the behavior of everything from ants to traffic to clouds to galaxies. Impractically speaking, it became a cult classic for those keen on wasting time. The spectacle of Life cells morphing on computer screens proved dangerously addictive for graduate students in math, physics and computer science, as well as for many people with jobs that provided access to idling mainframe computers. A U.S. military report estimated that the workplace hours lost clandestinely watching Life evolve on computer screens cost millions of dollars. Or so one Life legend has it. Another purports that when Life went viral in the early-to-mid-1970s, one-quarter of all the world’s computers were playing.

Courtesy of James Gardner, Martin Gardner Papers, Special Collections, Stanford University Libraries

A survey of Life-forms, from a 1970 letter to Martin Gardner.

Yet when Conway’s vanity strikes, as it often does, and he opens the index of a new mathematics book, casually checking for his name, he gets peeved that more often than not his name is cited only in reference to the Game of Life. Aside from Life, his myriad contributions to the canon run broad and deep, though with such meandering interests he considers himself quite shallow. There’s his first serious love, geometry, and by extension symmetry. He proved himself by discovering what’s sometimes called Conway’s constellation — three sporadic groups among a family of such groups in the ocean of mathematical symmetry. The biggest of his groups, called the Conway group, is based on the Leech lattice, which represents a dense packing of spheres in 24-dimensional space where each sphere touches 196,560 other spheres. He also shed light on the largest of all the sporadic groups, the Monster group, in the “Monstrous Moonshine” conjectures, reported in a paper composed frenetically with his eccentric Cambridge colleague Simon Norton. And his greatest masterpiece, in his own opinion at least, is the discovery of a new type of numbers, aptly named “surreal” numbers. The surreals are a souped-up continuum of numbers, including all the reals — integers, fractions and irrationals such as Euler’s number (2.718281828459045235360287471352662 … ) — and then going above and beyond and below and within, gathering in all the infinites, all the infinitesimals, and amounting to the largest possible extension of the real-number line. In Gardner’s reliable assessment, the surreals are “infinite classes of weird numbers never before seen by man.” And they may turn out to explain everything from the incomprehensible infinitude of the cosmos to the infinitely tiny minutiae of the quantum.

But the truly amazing thing about the surreal numbers is how Conway found them: by playing and analyzing games. Like an Escher tessellation of birds morphing into fish — focus on the white and you see the birds, focus on the red and you see fish — Conway beheld a game, such as Go, and saw that it embedded or contained something else entirely, the numbers. And when he found these numbers, he walked around in a white-hot daydream for weeks.

During his heyday at Cambridge in the 1970s, sandals-in-all-seasons Conway would typically saunter into the mathematics department common room and announce his arrival by slapping his hand on one of the large steel girders in the middle of the room. This generated a satisfyingly dissonant dinggggg. Another day of play now in session. One game, called Phutball, provided endless amusement.

### Phutball Rules

As described in the paper “Phutball Endgames Are Hard,” by Erik Demaine, Martin Demaine and David Eppstein: “John Conway’s game Phutball, also known as Philosopher’s Football, starts with a single black stone (the ball) placed at the center intersection of a rectangular grid such as a Go board. Two players sit on opposite sides of the board and take turns. On each turn, a player may either place a single white stone (a man) on any vacant intersection, or perform a sequence of jumps. To jump, the ball must be adjacent to one or more men. It is moved in a straight line (orthogonal or diagonal) to the first vacant intersection beyond the men, and the men so jumped are immediately removed. If a jump is performed, the same player may continue jumping as long as the ball continues to be adjacent to at least one man, or may end the turn at any point. Jumps are not obligatory: one can choose to place a man instead of jumping. The game is over when a jump sequence ends on or over the edge of the board closest to the opponent (the opponent’s goal line) at which point the player who performed the jumps wins. It is legal for a jump sequence to step onto but not over one’s own goal line. One of the interesting properties of Phutball is that any move could be played by either player, the only partiality in the game being the rule for determining the winner.”

Conway invented this game, a two-player board game with stones governed by wickedly negative feedback, with a Greek chorus of graduate students at his knee. But despite the fact that he made it up himself, this is not a game at which Conway excels.

Every time you take your turn you get this horrible feeling in the pit of your stomach. Because every move is bad. Instead of selecting the move that is best, you select the move that is least bad.… You make any move and immediately feel you shouldn’t have done it, and you think to yourself, Oh God, what have I done?

A de facto Phutball rule allows that if after a particularly excruciatingly bad move a player says, “Please, may I cry?” and the request is granted, then the move can be taken back and replayed. But even with such concessions, Conway is not very good at Phutball, and indeed he is not very good at game playing generally, or at least not very good at winning. Nevertheless, he was the perpetrator of endless gaming sessions in the common room, ultimately elevating games to a suitable subject for serious research, albeit punctuated by spasmodic outbursts in which he leapt into the air, latched onto a pipe along the ceiling, and swung violently back and forth.

Conway had good reason to worry about losing his job. By 1968, he had not accomplished much. All he was doing, after all, was squatting in the common room playing games, inventing games and reinventing rules to games he found boring.

Kelvin Brodie, The Sun News Syndication

John Conway playing the Game of Life in 1974.

Conway likes games that move in a flash. He used to play backgammon constantly, for small stakes — money, chalk, honor — though for all that practice he was not terribly good at backgammon, either. He took too many risks, accepting doubles when he shouldn’t and upping the ante to as much as 64 times the original stakes merely to see what would happen, all the while talking math. For example, there was Conway’s Piano Problem, which asked: What’s the largest object that can be maneuvered around a right-angle corner in a fixed-width corridor? (The lower bound for the object’s area is 2π + π2. It is possible to do better. But to find out how much better is very difficult.) He wasn’t interested in winning at backgammon as much as he was interested in the possibilities of the game. He liked to play a flamboyant “back game,” falling intentionally behind with inexplicably loony plays. Opponents, witnessing such folly, would let their guard down and get careless, gradually losing ground. Then Conway would make his move. Usually this strategy backfired and he lost as expected. But every now and then, depending on the luck of the dice — the element of chance is key in backgammon, and consequently the game defies much mathematical analysis and any pretensions of a serious research agenda — Conway would successfully rush in from behind and pull off a spectacular win.

While Conway was hopelessly addicted to backgammon, some of his colleagues carefully rationed their own participation, and others abstained outright, fearing that if they submitted at all they’d be sucked in and their research derailed. Other colleagues expressed concern that Conway was setting a bad example and corrupting the souls of graduate students. This, of course, was his plan.

One such student was Simon Norton, a child prodigy who had attended Eton College and managed to earn an undergraduate degree at the University of London during his last year of secondary school. When he arrived at Cambridge, Norton, already a backgammon whiz, easily fell in with the crowd. A lightning-fast calculator, he became Conway’s protégé, working out all the problems Conway couldn’t solve. He kept tabs on virtually all problems under way by everyone, snooping and eavesdropping and interrupting and bleating out “Fallllllssse!!” when he noticed a mistake. He also had a capacious vocabulary, which the logophile Conway appreciated, at least when Norton deigned to display this talent. He was known for his speedy solutions in games of anagrams that flew around the room in the interest of wasting time. To wit, one day someone served up “phoneboxes.” And before anyone could even cock his head to ponder, Norton declared: “Xenophobes!”

Mostly Conway played silly children’s games — Dots and Boxes, Fox and Geese — and sometimes he played them with children, primarily his four young girls. And of course he also played games with his floating population of acolytes, often games they invented for his delectation. Colin Vout came up with the game COL and Simon Norton made up SNORT, both map-coloring games. Norton also produced Tribulations, and Mike Guy parried with Fibulations, both Nim-like games based on triangle numbers and Fibonacci numbers. Conway invented Sylver Coinage, in which two players alternate in naming different positive integers, but they are not allowed to name any number that is the sum of any previously named number, and the first player who names “1” is the loser.

Many of these games went into the book Winning Ways for Your Mathematical Plays, by Conway and two co-authors, Elwyn Berlekamp, a mathematician at the University of California, Berkeley, and Richard Guy, a mathematician at the University of Calgary.

Courtesy of Elwyn Berlekamp

The game-theory trio of John Conway, Elwyn Berlekamp and Richard Guy came together during a “Computers in Number Theory” conference at the University of Oxford in 1969. Conway is on the third row from the top, second from right (with a beard). Berlekamp is in the fourth row from the top, sixth from right (also with a beard). Guy is in the fourth row from the top, ninth from the left (in a striped tie). Among other mathematical luminaries in the photo, the peripatetic mathematician Paul Erdős is seated in the front row, with the young girl in his lap.

The book took 15 years to write, in part because Conway and Guy were prone to silliness, punning back and forth and wasting Berlekamp’s time — Berlekamp called them “a couple of goons.” In the end and against all odds the book became a bestseller (the color printing and unusual typefaces increased the production costs so much that the advertising budget decreased to nothing). It was a self-help book, of sorts, on how to win at games. The authors spilled out a cornucopia of theories, along with many new games to match the theoretical purposes. According to Conway:

We would invent a new game in the morning with the intention of it serving as an application of a theory. And then after half an hour’s investigation, it would prove to be stupid. So we’d invent another game. There are 10 half-hours in the working day, roughly speaking, so we invented 10 games a day. We’d analyze them and sift them, and let’s say one in 10 of them was good enough to make the book.

They amassed a surfeit of games without names and names without games.

This was the Marriage Problem. You see we would invent a new game, and if it was a success, there would then be the problem of giving it some catchy kind of name. We’d try out a name, and usually we wouldn’t solve this naming problem. So the game might go in the file called “Games Without Names.” And then Richard, being his usual precise, persnickety self, had another file called “Names Without Games.” Any attempt to invent a new name for a game generated a whole lot of names, none of which were quite right, but they were quite often good names. So they went in the “Names Without Games” file. Each of these lists grew. And we seldom managed to marry one from this file to one from that file.

I remember the name without a game that was the best name without a game. It was called Don’t Ring Us, We’ll Ring You. We never got around to inventing that game, but the type of game is pretty clear: In this game there would be some thing or another that each player would draw on paper, and the aim is to draw a ring around your opponent. For a game like that, Don’t Ring Us, We’ll Ring You would be a lovely name. But we never actually found a game to fit it.

Every so often, Conway visited Martin Gardner and the two traded material on mathematical recreations — if not games, per se, then puzzles, and all sorts of nerdish delights. Take, for instance, Conway’s Doomsday Algorithm, by which he displayed his prodigious skill at naming the day of the week for any given date. Although Conway had been showing off this trick since he was a teen, the algorithm came about during a visit with Gardner. Conway flew to New York and waited for his friend to pick him up at the airport. And he waited, and waited, and waited. Gardner did not turn up as planned.

Initially I thought, Okay, he’s going to turn up in five minutes. But I waited there a hell of a long time, probably an hour, I don’t know. And I had started to think, “Well, what happens if he doesn’t turn up?” I didn’t have a phone number for him. And it wouldn’t matter if I did because I didn’t know how to work the American pay-phone system — I’m still like this, you might notice. So the easiest thing to do was to just sit there and hope.

More than two hours late, Gardner came running in, waving madly from the far end of the arrivals terminal, apologetic and promising, “You’ll forgive me as soon as you know what I’ve just discovered!” He’d been at the New York Public Library, where he had found a note published in an 1887 issue of Nature magazine — “To Find the Day of the Week for Any Given Date,” sent in by Lewis Carroll, who wrote: “Having hit upon the following method of mentally computing the day of the week for any given date, I send it you in the hope that it may interest some of your readers. I am not a rapid computer myself, and as I find my average time for doing any such question is about 20 seconds, I have little doubt that a rapid computer would not need 15.” Gardner couldn’t resist photocopying this choice find, but there was a long queue at the copy machine. He got in line. The line moved slowly. By the time it became apparent that he was bound to be late picking up Conway, he’d already invested 30 minutes, and he figured another 15 would suffice. He felt it was worth the wait, and he knew Conway would agree.

When they finally arrived at Gardner’s home, Gardner went straight to his file cabinets and produced 20-odd articles about working out the day of the week for any given date. The Lewis Carroll rule, in his view, was the best yet. All the same, he turned to Conway and said, “John, you ought to work out an even simpler rule that I can tell my readers.” And so during what Conway refers to as the long winter’s nights after Mr. and Mrs. Gardner had toddled off to bed (though the visits were always in the summer), Conway thought about how to work out the day of the week in a way he could explain to the average anyone on the street.

He was still thinking during the flight home and back in the common room, when he hit upon a method he called the Doomsday Rule. The algorithm requires only addition, subtraction and memory. Conway devised a mnemonic method of sorts, whereby as you work through the algorithm you store all the necessary information on the fingers of your outstretched hand — outstretched so as to better bear the burden of the megabytes. And in order to remember a certain important piece of information about the date in question, Conway bares his teeth and bites into his thumb really hard.

Tooth marks must be showing! That way the thumb remembers. And whenever I lecture on this I go to someone in the front row and ask them to certify that they can see the tooth marks. It really does help. You can’t get serious people to do it, because they think it is childish. But the point about doing it is that this whole business occupies quite a substantial part of your brain, and then you forget what the person said their birthday was. This way the thumb remembers how far the birthday was away from the nearest Doomsday, and your thumb is perfectly capable of remembering that for you.

Over the years Conway has taught the Doomsday Rule to thousands upon thousands of people — and on occasion as many as 600 or so at a time, all crammed together in a conference hall calculating each other’s birthdays and biting their thumbs. And always endeavoring to be unreasonable, Conway was not satisfied with his easiest of algorithms. As soon as he designed it, he started improving it — with some doggerel poetry (another mnemonic of sorts) composed by Richard Guy. His main motivation was that he yet again wanted the rule to be as simple as possible, especially for the purposes of teaching.

In addition to his regular visits, Conway had made a habit of summarizing his recreational research in lengthy letters to Gardner. He’d feed a hefty roll of foolscap, like butcher paper, into his typewriter, and type out an ongoing stream until it was long enough to send — three or four feet would be long enough, he figured, though Gardner cut up one letter into the equivalent of 11 legal-size pages.

Conway typically began his letters with a preamble:

I got your first parcel of books just before Christmas, and was so delighted I spent the next few days reading and re-reading them, particularly the Annotated Alice, which is superb. (My wife was very annoyed with you!)

Then he’d launch into research updates, beginning with, say, (1) his solution for dividing cake, then moving on to (2) a new wire and string puzzle, and then the bulk of the letter given over to:

3) Sprouts. The following game was invented a fortnight ago, on a Tuesday afternoon. By Wednesday it had infected our Maths dept beyond recall — even the secretarial staff had succumbed. We started with n spots on a piece of paper. The move is to join two of these spots — which are allowed to be the same spot — by a curve, and then to create a new spot on this curve. The curve must not pass through old spots, nor may it cross old curves, and at no time may any spot have more than 3 arcs emanating from it. In normal sprouts a player who cannot make a move loses, so that the object is to move last — in misère sprouts the last player loses.

Courtesy of James Gardner, Martin Gardner Papers, Special Collections, Stanford University Libraries

Sprouts, from a letter to Martin Gardner.

Sprouts, invented with his graduate student Mike Paterson, became the subject of a Scientific American column published in July 1967. Working on the column, Gardner wrote back to Conway with a list of questions, leaving more than ample space for him to fill in the answers, beginning with a question about his name, John H. Conway: “What does the H stand for?”

Horton. Why so much space for this? Did you expect something like Hog- ginthebottomtofflinghame-Frobisher-Williamss-Jenkinson?

Gardner also wanted more details on the genesis of the game. “I predict that it will become such a standard, well known game that it will be of interest to record a few details about the circumstances surrounding its invention,” Gardner wrote. “Could you supply a few details? Doodling during a lecture? (If so, what lecture?) Doodling over glasses of beer?”

We were doodling long after teatime in the Department common room trying to invent a good pencil and paper game. This was some days after I’d more or less completely analyzed the Lucasian game, an old game also with spots, but with no new spots added, so it doesn’t “sprout.” It originally came from a rather complicated game about folding stamps which [Mike Patterson] had put into pencil and paper form, and we were successively modifying the rules. At one point [Mike] said “why not put a new spot in the middle” … and as soon as this was adopted all the other rules were discarded, the starting position was simplified to just n points (originally 3), and sprouts sprouted. …

The day after sprouts sprouted it seemed that everyone was playing it. At coffee or tea times there were little groups of people poring over ridiculous-to-fantastic sprout positions. Some people were already attacking sprouts on Klein bottles and the like, with at least one man thinking of higher-dimensional versions … one found the remains of sprout games in the most unlikely places.

Whenever I try to acquaint somebody new to the game nowadays, it always seems that they’ve already heard of it by some devious route. Even my 3 and 4 year old daughters play it with each other, though I can usually beat them.

And Conway kept it coming, heading the next month’s letter:

IMPORTANT BREAKTHROUGH IN SPROUTOLOGY!

Courtesy of James Gardner, Martin Gardner Papers, Special Collections, Stanford University Libraries

Named sprouts, from a letter to Martin Gardner.

Today, Gardner’s prediction about continued interest in the game has proved correct. The World Game of Sprouts Association is “devoted to the discovery of sprouts reality” and to “a serious exploration of the game,” and holds an annual championship tournament online. “For humans only” is one of the rules, since extensive computer analysis of the game over the years inspired some to enter their computer programs in the tournament rather than themselves. Conway only recently learned of the World Game of Sprouts Association, but he has been well aware of computers playing the game. Computers were all the rage when he invented Sprouts, and they were a large part of his motivation.

I was distressed. Computers were being used to solve a number of open problems — computers could solve problems standing for 100 years. We wanted to invent a game that would be hard to analyze by computer.

Although it took a while, in the early 1990s a trio from Bell Labs and Carnegie Mellon University produced a paper documenting a “Computer Analysis of Sprouts,” analyzing the winning strategy for games with up to 11 spots. “Beyond n = 11 their program was unable to cope with sprouting complexity,” Gardner reported back to his readers. Decades later, a pair of French students wondered whether the 11-spot record was beatable. As a hobby, they developed software called GLOP — based on the French comic-strip character Pif le chien, who says “Glop” to express satisfaction. They produced a doctoral thesis on the subject, and they claimed to have solved Sprouts games with up to 44 dots. When Conway heard this he was somewhat curious, if incredulous.

I doubt that very much. They are basically saying they have done the impossible. If someone says they’ve invented a machine that can write a play worthy of Shakespeare, would you believe them? It’s just too complicated. If someone said they’d been having some success teaching pigs to fly…. Though if they were doing that over in the field behind the Institute [for Advanced Study in Princeton], I would like to take a look.

For one final sampling of Conway’s infinite gamesomeness, consider the game Traffic Jams, in which a fictitious country is represented by a triangular map and towns are represented by letters, all named after real towns in Wales — such as Aberystwyth, Oswestry, and:

Llanfairpwllgwyngyllgogerychwyrndrobwllllantysiliogogogoch.

One suspects that Conway designed this game solely to provide himself with an opportunity to offhandedly pronounce Llanfairpwllgwyngyllgogerychwyrndrobwllllantysiliogogogoch, a word he saw stretched out on a sign at said town’s railway station and on a sign in the town square. He observed that the two signs differed slightly, having 57 and 58 letters, respectively. The pertinent question regarding this game is: What move should the first player make?

Chaim Goodman-Strauss

In the game Traffic Jams, four players start from Aberystwyth (A), Dolgellau (D), Ffestiniog (F) and Merioneth (M). Players take turns sliding down one-way streets to the next town. The game ends when all players are stranded at the town of Conwy (C), from which there is no escape. Once all players are stranded, the next player meant to move loses.

All these games provided raw data when Conway’s surreal-number theory was in development. The perfect guinea pigs, the two key players, were his eldest daughters, Susie and Rosie, then about 7 and 8.

Serendipitously, during the surreals’ period of gestation and invention circa 1970, the British Go champion, Jon Diamond, was then a Cambridge math undergraduate. He founded the Cambridge Go Society, fueling a steady run of Go games in the common room. Diamond, now president of the British Go Association, doesn’t recall ever playing Conway. That’s probably because Conway rarely if ever actually played the game. He lurked nearby, stared at the board, and wondered why the move Diamond or his pal had just made was a good move or a bad move. Conway recalled:

They would discuss it as they played, and kibitzers were sitting around saying, “Why’d you make that stupid move?” And it looked just the same as all the good moves to me. I never understood Go. But I did understand that near the end of the game it broke up into a sum of games — within the big game there were a few smaller games in various regions of the board. So that provided the spur for me to work out the theory of sums of partizan [sic] games.

This spur, as if one was necessary, encouraged ever more gaming. Conway always carried the necessary ammunition on his person, the better to snare an unsuspecting opponent. And oddly enough in this pursuit he kept himself semi-organized with a leather games case well stocked with dice, checkers, a board, paper, pencils, maybe some rope, and always a few decks of cards. Card games and card tricks were his strong suit. His analysis of games with students, professors or visitors, or by himself, barefoot on the common room floor, evolved from single games to compound games, with players playing lots of games at once — sometimes, say, a game of chess and a game of Go as well as a game of Domineering — and deciding, one turn at a time, which game to make their move in. He filled his usual landslides of foolscap analyzing these games. Then, as he told a reporter from Discover magazine who came calling at Cambridge:

I had a fantastic surprise. I realized that there was an analogy between what I was writing down and the theory of real numbers. Then I looked at it and found it was much more than an analogy. It was the real numbers.

And much, much more, which fittingly became known as the surreal numbers — the largest possible expansion of the real-number line — named as such by the Stanford computer scientist Donald Knuth. And forever thereafter, Conway did not worry about the hard-to-please workaholic Professor Frank Adams and his ilk. Conway figured his big discovery, which originated from playing silly games, took the bite out of the serious mathematicians. Once he found the surreals (and in the same 12-month period, his “annus mirabilis,” he invented the Game of Life and discovered the Conway group), he mandated what he calls “the Vow.” “Thou shalt stop worrying and feeling guilty; thou shalt do whatever thou pleasest.” He surrendered to his peripatetic curiosity and followed wherever it went, whether toward recreation or research, or someplace altogether nonmathematical.

Gardner summed up the surreals theory as “Vintage Conway: profound, pathbreaking, disturbing, original, dazzling, witty and splattered with outrageous Carrollian wordplay.… Are these not trivial beginnings? Yes, but they provide a secure foundation on which Conway … carefully builds a vast and fantastic edifice.” But an edifice of what? Conway, in a paper titled “All Numbers, Great and Small,” concluded with a similar question:

Is the whole structure of any use?

“It is on the boundary between funny stuff and serious mathematics,” said the late Hungarian-American mathematician Paul Halmos. “Conway realizes it won’t be considered great, but he might still try to convince you that it is.” Quite to the contrary. Conway believes the surreals are great, and there’s no “might” about it. If anything, he is keenly disappointed that the surreals haven’t yet led to something greater.

Where does all this position him in mathematics’ ancient intellectual odyssey toward beauty and truth? Conway on occasion (when asked) sees himself as part of a marching band winding through the streets of time. Then again, unless asked, he rarely if ever stands back to situate himself within the enterprise as a whole. Others have tried. In this age of top-10 lists, the Observer, the world’s oldest Sunday newspaper, listed Conway in its pantheon of mathematicians whose discoveries have changed our world. But just try to discuss the Observer’s list, by the columnist Alex Bellos, with Conway, not to mention another list on which he recently found himself, by Clifford Pickover in his book Wonders of Numbers, which contains a chapter dedicated to “A Ranking of the 10 Most Influential Mathematicians Alive Today.” Allude to either, and he demurs with a vengeance:

It’s nice in one way. It really means that I might be one of the best-known mathematicians in the present day, and this is not quite the same as being the best. And it’s probably because of Life. But it’s embarrassing. Because people might think I’m behind it in some way. And I assure you I’m not. And it’s particularly embarrassing because at least one of those lists doesn’t include Archimedes and Newton.

In Conway’s view, Archimedes is the preeminent father of mathematics. It was Archimedes who first truly understood the real numbers, and he was the first mathematician to work out the value of π, proving it was between the upper bound of 3 17; and the lower bound of 3 1071. Yet in the Observer’s ranking, it’s not Archimedes but Pythagoras at the top. If not the best mathematician, Pythagoras is perhaps the best-known, due to his namesake theorem. And generally the list comprises last-name-basis mathematicians who, in their day, appeared in the society pages of science: Euler, Gauss, Cantor, Erdős. Conway comes in toward the end, followed by Perelman and Tao, both of whom have been in the news lately. The Russian Grigori Perelman solved the Poincaré conjecture and refused all the accolades, including the Fields Medal. Terence Tao, a mathematician at the University of California, Los Angeles, is an expert in prime numbers who accepted his 2006 Fields Medal and in 2014 won the inaugural $3 million Breakthrough Prize in Mathematics. Conway’s salad days spanned the Sexy ’70s and the Excessive ’80s — and in the 1980s he divorced his first wife Eileen, married a mathematician named Larissa Queen and started another family; he became a Fellow of the Royal Society, and a full professor at Cambridge; and then he jumped ship to Princeton in 1987. With Perelman and Tao and even Conway, we are too close to evaluate the long horizon of their contributions, especially by the criterion of whether their pure and abstract math will evolve to find practical application. The verdict on that often takes time, sometimes a long time. The notable exception is the late John Nash, a colleague of Conway’s at Princeton and the subject of the book and movie A Beautiful Mind. Nash made contributions in game theory, and these were quickly put to use in evolutionary biology, accounting, politics, military theory and market economics, earning him a Nobel Memorial Prize in Economic Sciences. (In Conway’s view, Nash’s Nobel work is less interesting than the deep and difficult, albeit less useful, Nash embedding theorem, which states that every Riemann manifold can be isometrically embedded in Euclidean space.) Conway has been in the running for the million-dollar “Nobel” of mathematics, the Abel Prize — which is to say he’s been nominated, and the nomination remains on file — with his group-theory work being the strongest point in his favor. He has won other big math prizes, but as yet has had no luck with the Abel. And for the most part any practical implications of his work also remain to be seen. Few doubt that at least some of his gems will find application. The surreals, for instance. “The surreal numbers will be applied,” said his colleague, Peter Sarnak, a mathematician at the Institute for Advanced Study in Princeton. “It’s just a question of how, and when.” And Sarnak is one to sing Conway’s praises generally. “Conway is a seducer, the seducer,” he said, speaking exclusively of Conway’s skills as a teacher and expositor, of course — whether in the classroom, or at math camp, doing standing-room-only public lectures or private parties, or in his edifying alcove in the Princeton common room. He can always be found ensconced in his alcove, not working. He hasn’t given up all hope for hitting upon more white-hot math like the surreals, but more often than not he is “thinkering” away with his beloved trivialities. Conway has no compunction about buttonholing strangers and serving them a rollicking riff on his many obsessions. One obsession of late is the Free Will Theorem, in which, he points out, every human being has a vested interest. Devised over the course of a decade with his Princeton colleague Simon Kochen, the Free Will Theorem is precisely formulated using geometry, quantum mechanics and philosophy, though the duo usually state it very basically as follows: If physicists have free will while performing experiments, then elementary particles possess free will as well. And this, they reckon, probably explains why and how humans have free will in the first place. It isn’t a circular argument so much as a spiral argument, a self-subsuming argument, spiraling outward and getting bigger and bigger. But usually it’s numbers that are the object of his infatuation. He turns numbers over, upside down, and inside out, observing how they behave. Above all he loves knowledge, and he seeks to know everything about the universe. Conway’s charisma lies in his desire to share his incurable lust for learning, to spread the contagion and the romance. He is dogged and undaunted in explaining the inexplicable, and even when the inexplicable remains so, he leaves his audience elevated, fortified by the failed attempt and feeling somehow in cahoots, privy to the inside dope, satisfied at having flirted with a glimmer of understanding. This article was reprinted on Wired.com. View Reader Comments (9) Leave a Comment ## Reader CommentsLeave a Comment • Noel Gardeli says: This was an amazing read about a very interesting person. Thank you for it. It's nice how many of the greatest geniuses in history were like him (at least in the field of philosophy), in that they "just did what they wanted and when they wanted" (with exceptions sometimes, by need…), working for fun, in things they were curious about, they have produced many nice ideas: this may be one of the best "methods" of work (although the results usually come later than by other "methods"-in the case of philosophers only after the age of 40!) 😀 By the way, the best list of "top mathematicians" in history that I know (of course no one is able to rank geniuses objectively, but it seems very good and with good biographical informations) is the one by James Dow Allen, which can be found on http://fabpedigree.com/james/mathmen.htm. Notice how Newton and Archimedes are #1 and #2 respectively, but perhaps Archimedes should be #1. Conway is there too, at #107 (which is not bad! –the names there are only of geniuses!) • Portgawa mlys says: Well, recently i have seen something similar to Elementary Cellular Automata & conway game of life … when you create something in conway's game of life, you can create any structure, here (https://www.reddit.com/r/cellular_automata/comments/3hlaqz/how_would_one_describe_the_rules_of_pascals/cu8nve2) a user also describes that by somehow there is "infinite universe" within ours by using numbers and math operations. It took me some time to understand what is written in the url, but now it is clearly that there is a universe does exist with all the structures from elementary cellular automata, Pascal's Triangle alike-shapes & even patterns from conway's game of life ! I like the word (a triangle atom) mention by the user, it describes the process by somehow. it is suprising that no one mentioned it yet, the ability to build conway's game of life structures with usage of numbers only. • Deryk Barker says: Conway was my supervisor at Cambridge in the late '60s and I've long thought hi to be the only true genius I've ever met. I can still remember his telling me about Life and the$1 million computer time supposedly wasted in the year before Also his showing me the multiplication table for his group, a fanfold listing which spanned the four walls of his office in the maths faculty building. When I expressed the though that it was rather small for the size of the group, he said: "ah, but every symbol represents a 100 square matrix"…

I would suggest that, when I knew him, he was a lousy teacher though: he could never understand why anyone found anything mathematical difficult…

• Patrick Hayes says:

While an undergraduate at Cambridge in 1966 I sat in on a weekly postgrad seminar Conway was teaching on axiomatic set theory. The experience was exhilarating but I was struggling to keep up with the rather rapid pace. We had got some way into the transfinite hierarchy when one morning Conway showed up in a state of great excitement and told us that he had invented a new kind of infinite number larger than any other infinite number, and he would tell us the basic ideas at the nest meeting. A week later we all showed up eager to hear about it, but Conway never appeared. The following week he re-appeared, apparently back in the finite real world, but never mentioned the topic nor the reason for his disappearance, and none of us had the temerity to ask.
By the way, I disagree with Deryk. I found Conway to be a brilliant and inspiring teacher.

• Colin P says:

I was not yet a teenager through the 1960's, my dad used to get Scientific American and I can remember after we read about Conway's Game of Life spending hours as a lad drawing his cellular automata on sheets of dad's graph paper, especially the "gliders" but also many other shapes, fascinated to see what happened! Happy memories!

• Michael Gage says:

John Horton Conway was my supervisor at Gonville & Caius, Cambridge, 1961-, and I well remember his interest in whimsy. We used to ask what he was working on and it was no surprise when it was 5-D kettles, or suchlike. He also had a book illustrated with the strange beings that inhabited each of the planets. Those were the days!

• Thanks to Conway's Game of Life cellular automaton, I spend hundreds of hours of my teen years ruining graph paper and moving Lego blocks around on a gridded cloth my mom made. Then I built a computer using 7400-series TTL logic chips and a couple 2102 RAM chips (1024 bits each!) to do the logic automatically. It won recognition at the Detroit Metro Science Fair. Ah, fun times long ago…

• Charles Stromeyer Jr. says:

Thank you for writing this great piece about a mathematical genius.

I would like to add the observation that what Conway and Kochen call "free will" can be substituted for "free randomness", as explained in this paper by Roger Colbeck and Renato Renner published in "Nature Physics" which shows that 'free randomness' or 'free will' can be amplified (assuming a certain no-signalling theorem): http://arxiv.org/pdf/1105.3195.pdf

• Guillermo says:

Beautiful writing!

• Barney says:

Monstrous Moonshine is a great example of Conway's special kind of genius. It looks like a dumb idea (hence the name) but that never bothered Conway. And its consequences are vast.

Two other topics worth mentioning are Conway's work on knots, which is practically impossible to follow, and the look and say sequence. Also he co-wrote a wonderful book about symmetry which you will understand and enjoy the first few chapters of before realizing you're in way over your head.